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Mirrors > Home > MPE Home > Th. List > Mathboxes > un0.1 | Structured version Visualization version GIF version |
Description: ⊤ is the constant true, a tautology (see df-tru 1536). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
un0.1.1 | ⊢ ( ⊤ ▶ 𝜑 ) |
un0.1.2 | ⊢ ( 𝜓 ▶ 𝜒 ) |
un0.1.3 | ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) |
Ref | Expression |
---|---|
un0.1 | ⊢ ( 𝜓 ▶ 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0.1.1 | . . . 4 ⊢ ( ⊤ ▶ 𝜑 ) | |
2 | 1 | in1 44154 | . . 3 ⊢ (⊤ → 𝜑) |
3 | un0.1.2 | . . . 4 ⊢ ( 𝜓 ▶ 𝜒 ) | |
4 | 3 | in1 44154 | . . 3 ⊢ (𝜓 → 𝜒) |
5 | un0.1.3 | . . . 4 ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) | |
6 | 5 | dfvd2ani 44166 | . . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃) |
7 | 2, 4, 6 | uun0.1 44361 | . 2 ⊢ (𝜓 → 𝜃) |
8 | 7 | dfvd1ir 44156 | 1 ⊢ ( 𝜓 ▶ 𝜃 ) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1534 ( wvd1 44152 ( wvhc2 44163 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-vd1 44153 df-vhc2 44164 |
This theorem is referenced by: sspwimpVD 44502 |
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